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Magnetic Flux of a Rotating Ring

Patrick Ford
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Alright, guys, let's work this one out together. So we're told that you have a radius of some ring of some radius that is in the presence of a magnetic field. And the bring begins with its plane parallel to the magnetic field and ends perpendicular. We're gonna talk about the just in just a second we're supposed to be figuring out is what is the change in the magnetic flux. Now, the key word in this problem is the change. So in other words, we're looking for some magnetic flux. So that's fi B, but we're looking for the change and were represented by a Delta symbol, remember? So that means that we just need to find out what thief I be final is minus whatever fi be initial is. So we're basically gonna have to separate sort of initial and final states. We're gonna have to figure out those two. So the initial and the final I'm gonna go ahead and draw diagrams for those two right here. So let's say the magnetic field just happens to point to the right. It doesn't really matter where we choose to draw it. As long as we're just consistent, right So we have the magnetic field that points in this direction, and I'm just gonna assume that points to the right like this. We have the exact same magnetic field over here. Now, what happens in the between the initial and the final is that the ring is gonna rotate like this. And we're told that this ring begins with the plane parallel to the magnetic field. Now, what that means is that the actual ring itself the lines of the ring, the plane of it is parallel to the magnetic field, not the normal. And then what happens is finally it ends up with the plane of the ring perpendicular to the magnetic field. So that means it's actually going to be vertical like this. We're supposed to figure out what the change in magnetic flux is. All right, so we have our five b final are five the initial, so we can go ahead and write out the equations. For those that's Delta Phi B is equal to this is gonna be be a and then we have the co sign of feta. What happens is we know that the angle is changing, whereas B and A are going to remain constant. So these guys don't actually change these air constant right here. So let me write that. So these are constants, and what's actually happening is that we have an angle so theta final and then minus b a times cosine of theta initial. And these angles right here represent the angle between B and A. So let's go ahead and find out what those are. Let's start with the final case. Right. So we know that if the ring sort of sits vertically, that means the plane of it is gonna be or sort of the normal is gonna be perpendicular to that surface. So that means it's gonna point out the right there's always wanted on appointing alongside the magnetic field. So what's the angle between B and A. Well, these things points in the same direction. That means that this theta here, which is actually fate a final, is gonna be equal to zero degrees, right, because they point in the same direction. And so the co sign of zero degrees is just equal to one Now, what that does for our equation is we say Okay, well, if co sign of zeros, one That means this term right here just equals one. And we can sort of just get it out of there because one doesn't do anything to the equation. So that means this is actually just equal to B times A. Now we have to look at what the initial angle is. We have a minus sign right here. Right. So let's look at what the initial angle is. Well, the area is again perpendicular to the surface, so it's gonna be pointing up in this case because now we have the ring sort of lying flat like that, and it's always gonna be perpendicular. So this is our area Vector and theta angle represents the area between B and A. So that's actually gonna be this guy right here. Now, hopefully, you guys realize that this angle right here, feta initial is actually equal to 90 degrees. And if you go ahead and work out what the co sign of 90 degrees in your calculator is, you're gonna get zero. So what happens is this whole entire term gets wiped out because this is just equal to zero. Okay, so that means that the change in the magnetic field is literally just be times a and then minus zero because we have nothing there. Let's just go ahead. Multiply those two out. So we have a 20.6 Tesla magnetic field, and then the radius is two centimeters so that we have We have pie times 20.2 squared. And so you work this out, you're gonna get 7.54 times 10 to the minus four, and that's gonna be Weber's. Okay, so that's the change in the magnetic field. Let me know if you guys have any questions and I'll see you in the next one.